144
Orlicz-Morrey
spaces and
some
integral operators
大阪教育大学教育学部 中井英一 (Eiichi Nakai)
Department of Mathematics
Osaka Kyoiku University
[email protected] 1. INTRODUCTION
For the Hardy-Littlewood maximal operator $M$, Calder\’on-Zygmund operator $T$
and fractional integral operator Ia, $0<\alpha<n,$ it is well known that
$M$ : $L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}(\mathbb{R}^{n})$ , $M$ : $L^{p}(\mathbb{R}^{n})arrow L^{p}(\mathbb{R}^{n})$, $1<p\leq\infty$,
$T$ : $L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}($’$n)$, $T:L^{\mathrm{p}}(\mathbb{R}^{n})arrow L^{p}(\mathbb{R}^{n})$, $1<p<\infty$,
$I_{\alpha}$ : $L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{n/(n-\alpha)}(\mathbb{R}^{n})$, $I_{\alpha}$ : $L^{p}(\mathbb{R}^{n})$ $arrow L^{q}(\mathbb{R}^{n})$, $\{$
$1<p<q<\infty$,
$-n/p+\alpha=-n/q$,
Theseboundedness extended toseveralfunction spaces whichare generalizations of
$L^{p}$ZAspaces, for example, Orlicz spaces, Morrey spaces, Lorentz spaces, Herz spaces,
etc.
For boundedness of$M$
on
Orlicz spaces $L^{\Phi}(\mathbb{R}^{n})$, Kita [10] (1997) proveda
nec-essary and sufficient condition on 4 and $\Psi$ for
$M$ : $L^{\Phi}(\mathbb{R}^{n})arrow L^{\Psi}(\mathbb{R}^{n})$.
Cianchi [3] (1999) proved necessary and sufficient conditions on (1) and I for
$M$,$T$,$I_{\alpha}$ : $L^{\Phi}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Psi}(\mathbb{R}^{n})$,
$M$,$T$,$I_{\alpha}$ : $L^{\Phi}(\mathbb{R}^{n})arrow L^{\Psi}(\mathbb{R}^{n})$
.
For boundedness of$I_{\alpha}$on Morreyspaces$IP^{\lambda}’(\mathbb{R}^{n})$, seePeetre (Spanne) [18] (1969)
Adams [1], 1975 Chiarenza and Frasca [2] (1987), Nakai (1995). Chiarenza and
Frasca showed
$T$ : $L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}(\mathbb{R}^{n})$ , $T:L^{\mathrm{p}}(\mathbb{R}^{n})arrow L^{p}(\mathbb{R}^{n})$, $1<p<\infty$,
$I_{\alpha}$ : $L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{n/(n-\alpha)}(\mathbb{R}^{n})$, $I_{\alpha}$ : $L^{p}(\mathbb{R}^{n})arrow L^{q}(\mathbb{R}^{n})$, $\{_{-n/p+\alpha=-n/q}1<p<q<\infty$,
,
Theseboundedness extended toseveralfunction spaces whichare generalizations of
$L^{p}$ZAspaces, for example, Orlicz spaces, Morrey spaces, Lorentz spaces, Herz spaces,
etc.
For boundedness of$M$
on
Orlicz spaces $L^{\Phi}(\mathbb{R}^{n})$, Kita [10] (1997) proveda
nec-essary and sufficient condition on $\Phi$ and $\Psi$ for
$M$ : $L^{\varphi}(\mathbb{R}^{n})arrow L^{\mathrm{u}}\mathrm{r}(\mathbb{R}^{n})$.
Cianchi [3] (1999) proved necessary and sufficient conditions on $\Phi$ and $\Psi$ for
$M$,$T$,$I_{\alpha}$ : $L^{\Phi}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Psi}(\mathbb{R}^{n})$,
$M$,$T$,$I_{\alpha}$ : $L^{\Phi}(\mathbb{R}^{n})arrow L^{\Psi}(\mathbb{R}^{n})$
.
For boundedness of$I_{\alpha}$on Morreyspaces$L^{p,\lambda}(\mathbb{R}^{n})$, seePeetre (Spanne) [18] (1969)
Adams [1], 1975 Chiarenza and Frasca [2] (1987), Nakai (1995). Chiarenza and
Frasca showed
$M,T:L^{p,\lambda}(\mathbb{R}^{n})arrow L^{p,\lambda}(\mathbb{R}^{n})$,
$I_{\alpha}$ : $L^{p,\lambda}(\mathbb{R}^{n})arrow L^{q,\lambda}(\mathbb{R}^{n})$
.
The author studied boundedness ofgenaralized fractional integral operators
on
Orlicz-Morrey spaces in [17]. Orlicz-Morrey spaces
are
useful to estimategenar-alized fractional integral operators. In this paper we investigate boundedness and
145
weak boundedness of the Hardy-Littlewood maximal operator, singular integtral
operators and generalized fractional integral operators. We refine on the results in [17].
2. DEFNITIONS
A function $\theta$ :
$(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$ is said to be almost increasing (almost
de-creasing) ifthere exists a constant $C>0$ such that
$\theta(r)\leq C\theta(s)$ $(\theta(r)\geq C\theta(s))$ for $r\leq s.$ A function $\theta$ : $(0, +\mathrm{o}\mathrm{o})arrow$
$(0, +\mathrm{o}\mathrm{o})$ is said to satisfy the doubling condition if there
exists a constant $C>0$ such that
$C^{-1} \leq\frac{\theta(r)}{\theta(s)}\leq c$ for $\frac{1}{2}\leq\frac{r}{s}\leq 2.$
For functions 0,$\kappa$ : $(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$, we denote
$\theta(r)\sim\kappa(r)(\mathrm{O}(\mathrm{r})\approx\kappa(r))$ if
there exists
a
constant $C>0$ such that$C^{-1}/1/(r)$ $\leq\kappa(r)\leq C\theta(r)$ $(\theta(C^{-1}r)\leq\kappa(r)\leq\theta(Cr))$ for $r>0.$
Let$\mathcal{Y}$be the set ofallconvexfunctions(I) :
$[0, +00)arrow[0, +\mathrm{o}\mathrm{o})$ such that $\Phi(r)>0$ for $r>0$
.
If$\Phi\in$ )), then $\Phi$ is increasing, absolutely continuous and bijective.For
a
measurable set $\Omega\subset \mathbb{R}^{n}$, we denote the characteristicfunction of $\Omega$ by
$\chi_{\Omega}$
and the Lebsgue
measure
of0 by $|\Omega|$.
Fora
measuralbe set $\Omega\subset \mathbb{R}^{n}$,a
measuralbefunction $f$ and $t>0,$ let
$m(\Omega, f, t)=|\{x\in\Omega : |f(x)|>t\}|$
.
In the
case
$\Omega=\mathbb{R}^{n}$,we
shortly denote it by$m(f, t)$
.
Definition 2.1 (Orlicz space). For $\Phi$ $\in \mathcal{Y}$ let
$L^{\Phi}(\mathbb{R}^{n})=$ $\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n}) :||f||_{L^{\Phi}}<+\mathrm{o}\mathrm{o}\}$,
$||f||_{L^{\Phi}}= \inf\{\lambda>0:\int_{\mathrm{R}^{n}}\Phi(\frac{|f(x)|}{\lambda})dx\leq 1\}$ ,
$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})=\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n})$ :
$||f||\mathrm{z}\mathrm{H}\mathrm{e}*\mathrm{k}$ $<+\mathrm{o}\mathrm{o}\}$ , $||f11_{L\mathrm{Z}\epsilon \mathrm{U}}$ $=$ $\inf$ $\{\lambda>0$ : $\sup_{\mathrm{t}>0}\Phi(t)m(\frac{f}{\lambda},$$t)\leq 1\}$
If$\Phi(r)=r^{p}$, $1\leq p<\infty$, then $L^{\Phi}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$ and $L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})=L^{p}$
y
$\mathrm{e}\mathrm{a}\mathrm{k}($’$n)$.If$\Phi(r)\leq\Psi(Cr)$, then $L^{\Phi}(\mathbb{R}^{n})\supset L^{\Psi}(\mathbb{R}^{n})$ and $||f||_{L^{\Phi}}\leq C||f||_{L^{\Psi}}$
.
If$\Phi\approx\Psi$, then $L^{\Phi}(\mathbb{R}^{n})=L^{\Psi}(\mathbb{R}^{n})$ and $||f||_{L^{\Phi}}\sim||f||_{L}\mathrm{v}$.
14El
Let $B(a, r)$ be the ball $\{x\in \mathbb{R}^{n} : |x-a\mathit{9}|<r\}$ with center $a$ and of radius $r>0.$
Definition 2.2 (Morrey space). For $1\leq p<\infty$ and $0\leq$ X $\leq n,$ let
$L^{p,\lambda}(\mathbb{R}^{n})=$ $\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n}) :||f||L\mathrm{p},’ <+\mathrm{o}\mathrm{o}\}$,
$||f||_{L^{\mathrm{p},\lambda}}= \sup_{B=B(a,r)}$
(
$\frac{1}{r^{\lambda}}\int_{B}|f(x)|p$$dx$
),
$L\mathrm{w}\mathrm{e}_{\mathrm{a}\mathrm{k}}(’ n)$ $=\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n})$ : $||f||L\mathrm{p}\mathrm{w}’ ex\mathrm{k}<+\infty\}$ ,
$||f||_{L_{\mathrm{w}\epsilon \mathrm{a}\mathrm{k}}^{\mathrm{p}.\lambda}}= \sup_{B=B(a,t)}\sup_{t>0}\frac{t^{p}m(B,f,t)}{r^{\lambda}}$
.
If A $=0,$ then $U^{\lambda},(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$ and $L_{\mathrm{w}\acute{\mathrm{e}}\mathrm{a}\mathrm{k}}^{p\lambda}(\mathbb{R}^{n})=L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{p}(\mathbb{R}^{n})$ . If A
$=n,$ then
$L^{p,\lambda}(\mathbb{R}^{n})=L^{\infty}(\mathbb{R}^{n})$.
Let $\mathcal{G}$ be the set of all functions $\phi$ : $(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$ such that 6 is almost
decreasing and $\phi(r)r^{n}$ is almost increasing. If $\phi\in \mathrm{C}\mathrm{i}$ then 6 satisfies doubling
condition.
Definition 2.3. For $1\leq p<\infty$ and
a
function $\phi\in \mathcal{G}$,$L^{(p,\phi)}(\mathbb{R}^{n})=$ $\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n}) :||f||_{L}(\mathrm{p},\phi)<+\mathrm{o}\mathrm{o}\}$,
$||f||_{L^{(\mathrm{p},\phi)}}$ $=\mathrm{s}B=\mathrm{u}_{(\mathrm{a},\mathrm{r})}^{\mathrm{P}}$ $( \frac{1}{|B|\phi(r)}\int_{B}|f(x)|^{p}dx)^{1/p}$ , $L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(p,\phi)}(\mathbb{R}^{n})=\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n})$: $||f||_{L}$
Pe&2
$<+\mathrm{o}\mathrm{o}\}$, $||f||_{L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\mathrm{p},\phi)}}= \sup_{B=B(a,r)}\sup_{t>0}\frac{t^{p}m(B,f,t)}{|B|\phi(r)}$.
If $\phi(r)=r^{-n+}$
’,
$0\leq$ A $\leq n,$ then $L^{(p,\phi)}(\mathbb{R}^{n})=L^{p,\lambda}(\mathbb{R}^{n})$ and $L_{\mathrm{w}\acute{\mathrm{e}}\mathrm{a}\mathrm{k}}^{(p\phi)}(\mathbb{R}^{n})=$$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{p,\lambda}(\mathbb{R}^{n})$
.
If$p\leq q,$ then $L^{(p,\phi)}(\mathbb{R}^{n})\supset L^{(q,\phi^{q/\mathrm{p}})}(\mathbb{R}^{n})$ and $||f||_{L(\mathrm{p},\phi)}\leq||f||_{L^{(q}}$
,4q/p). If $\phi(r)\leq$ $C\psi(r)$, then $L^{(p,\phi)}(\mathbb{R}^{n})\subset L^{(p,\psi)}(\mathbb{R}^{n})$ and $||f||_{L(\mathrm{p},\phi)}$ $\mathit{2}$ $C^{-1/p}||f||_{L(\mathrm{p},\psi)}$
.
If $6\sim\psi$, then$L^{(\mathrm{p},\phi)}(\mathbb{R}^{n})=L^{(p,\psi)}(\mathbb{R}^{n})$ and $||f||_{L^{(\mathrm{p},\phi)}}$ $\sim||f||_{L^{(\mathrm{p},\psi)}}$
.
For $\Phi\in$ )),
a
function $\phi\in(\mathrm{i}$ anda
ball $B=B(a, r)$, let$|\mathrm{V}||$
,,\phi ,B $=$ inf$\{$A $>0$ : $\frac{1}{|B|\phi(r)}\int_{B}\Phi(\frac{|f(x)|}{\lambda})dx\leq 1\}$,
$||f||_{\Phi,\phi,B,\mathrm{w}\mathrm{e}}$
147
We note that
$\sup_{t>0}\Phi(t)m(\Omega, f, t)=\sup_{t>0}tm(\Omega, f, \mathrm{I}^{-1}(t))=\sup_{t>0}tm(\Omega, \Phi(17|)$,$t)$.
Definition 2.4 (Orlicz-Morrey space). For $\Phi\in$ )) and a function $\phi\in \mathcal{G}$, let
$L^{(\Phi,\phi)}(\mathbb{R}^{n})=$ $\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n}) :|| 7 ||_{L}(\Phi,\phi)<+\mathrm{o}\mathrm{o}\}$
: $||f||_{L^{(\Phi,\phi)}}$ $= \sup_{B}||f||_{\Phi}$,$\phi$,B, $L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\Phi,\phi)}(\mathbb{R}^{n})=\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n})$: $||f||L\mathrm{y},\mathrm{z}$) $<+$oo$\}$ , $||f||L\mathrm{r}’ \mathrm{b})$ $=\mathrm{s}\mathrm{u}^{\mathrm{P}}$ $||f||_{\Phi,\phi,B,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}$
.
If$\Phi \mathrm{z}$ $\Psi$ and $6\sim\psi$, then $L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{(\Psi,\psi)}(\mathbb{R}^{n})$.
If$\Phi(r)=r^{p}$, then $L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{(p,\phi)}(\mathbb{R}^{n})$ and $L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\Phi,\phi)}(\mathbb{R}^{n})=L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(p,\phi)}(\mathbb{R}^{n})$. If
$\phi(r)=$ $r^{-n}$, then $L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{\Phi}(\mathbb{R}^{n})$ and $L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\Phi,\phi)}(\mathbb{R}^{n})=L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})$
.
If $\phi(r)\equiv 1,$ then$L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{\infty}(\mathbb{R}^{n})$. If$\Phi(r)$ $=r^{p}$ and $\phi(r)=r"-n$, then $L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{p,\lambda}(\mathbb{R}^{n})$
.
A function $\Phi\in$ $\mathcal{Y}$ is said to satisfy the $\Delta_{2}$-condition, denoted $\Phi\in\Delta_{2}$, if
$\Phi(2r)\leq C\Phi(r)$, $r\geq 0,$
for
some
$C>0.$A function $\Phi\in$ $\mathcal{Y}$ is said to satisfy the $\nabla_{2}$-condition, denoted $\mathrm{I}$
$\in 7_{2}$, if
$\Phi(r)\leq\frac{1}{2k}\Phi(kr)$, $r\geq 0,$
for
some
$k>1.$Let $y+$ be the set of all (It $\in \mathcal{Y}$ with $\mathrm{r}\mathrm{o}^{1}\Phi(t)t^{-2}dt<+\mathrm{o}\mathrm{o}$
.
For (I) $\in J\mathit{1}^{+}$, let$\Phi^{+}(r)$ $=r \int_{0}^{f}\frac{\Phi(t)}{t^{2}}dt$, $r\geq 0.$
Then $\Phi^{+}\in$ ) and $\Phi(r)\leq\Phi^{+}(2r)$. If 4 satisfies the $\nabla_{2}$-condition, then $\Phi\in y+$
and $\Phi^{+}\approx\Phi$
.
Example 2.1. For $\epsilon>0$ and
a
$\geq 0,$ let $\Phi\in \mathcal{Y}$with$\Phi(r)=\{$
$r(\log(1/r))^{-\epsilon-1}$ for small $r>0,$
$r(\log r)^{\delta}$ for large $r>0.$
Then $\Phi\in$ $\mathcal{Y}+$ a $\mathrm{d}$
$\Phi^{+}(r)\approx\{$
$r(\log(1/r))^{-\epsilon}$ for small $r>0,$
148
Example 2.2. For $1<p<\infty$, $\epsilon\in \mathbb{R}$ and $\delta\in \mathbb{R}$, let $\Phi\in \mathcal{Y}$ with
$\Phi(r)=\{$
$r^{p}(\log(1/r))^{-\epsilon}$ for small $r>0,$
$r^{p}(\log r)^{\delta}$ for large $r>0.$
Then $\Phi\in\nabla_{2}$ and $\Phi^{+}\approx\Phi$
.
For afunction $\theta$ : $(0, +00)arrow(0,$$+\mathrm{o}\mathrm{o}$ , let
$\theta^{*}(r)=\int_{0}^{f}\frac{\theta(t)}{t}dt$, $\theta_{*}(r)=\int_{f}^{+oo}$ $\frac{\theta(t)}{t}dt$
.
If 0 satisfies the doubling condition, then $\theta(r)\leq C\theta^{*}(r)$ and $\theta(r)\leq C\theta_{*}(r)$
.
If$\theta(r)r^{-\epsilon}$ is almost increasing for
some
$\epsilon>0,$ then $\theta^{*}(r)\leq C\theta(r)$.
If$\theta(r)r^{\epsilon}$ is almostdereasing for
some
$\epsilon>0,$ then $\theta_{*}(r)\leq C\theta(r)$.
If $6\in \mathcal{G}$ and $7_{0}^{1}6(t)dt<+\mathrm{o}\mathrm{o}$, then $\phi^{*}\in \mathcal{G}$
.
If $\phi\in$ $\mathcal{G}$ and $\int_{1}^{+\infty}\phi(t)dt<+\mathrm{o}\mathrm{o}$,then $\phi_{*}$
:
$\mathcal{G}$.
3. MAIN RESULTS
The Hardy-Littlewood maximal function of $f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n})$ is defined by
$Mf(x)= \sup_{B\ni x}|\mathrm{g}|\int_{B}|f(y)|dy$,
where the supremum is taken
over
all balls $B$ containing $x$.The following
are our
main results.Theorem 3.1. Let $\phi$ $\in$ (i and $\Phi\in$ )$)$
.
Then$M$ : $L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Phi\phi)}(\mathbb{R}^{n})$
.
If
(I) $\in))^{+}$, then$M$ : $L^{(\Phi^{+},\phi)}(\mathbb{R}^{n})arrow L(")(\mathbb{R}^{n})$
.
If
$\Phi\in\nabla_{2}$, then$M$ : $L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Phi,\phi)}(\mathbb{R}^{n})$
.
Next let $T$ be
a
singular integral operator with a kernel $K(x, y)$ which satisfies(3.1) $|K(x, y)| \leq\frac{C}{|x-y|^{n}}$, $x$,$y\in \mathbb{R}^{n}$,$x\neq y,$ and, for $f\in C_{\mathrm{c}}$
o
$\mathrm{m}\mathrm{p}($’$n)$,148
Theorem 3.2. Let$\Phi\in$ $\Delta_{2}$, $\phi$,$\psi\in$ (; and $\mathit{7}_{1}^{+\infty}\Phi^{-1}(\phi(t))/tdt<+\mathrm{o}\mathrm{o}$. Assume that
there exists a constant $A>0$ such that
(i) $((\Phi^{-1}0\phi)_{*}(r))\leq A\psi(r)$, r $>0.$
(i)
If
$T$ : $L^{\Phi}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})$ , then$T:L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Phi\psi)}(\mathbb{R}^{n})$.
(ii)
If
$T:L^{\Phi}(\mathbb{R}^{n})arrow L^{\Phi}(\mathbb{R}^{n})$, then$T:L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Phi,\psi)}(\mathbb{R}^{n})$
.
Remark 3.1. Since $\Phi^{-1}\circ\phi$ satisfies doubling condition, $\phi(r)=$ $1$$(!-1\circ\phi(r))$ $\leq$
$\Phi(C(\Phi^{-1}\circ\phi)_{*}(r))\leq C’\psi(r)$ for all $r>0.$ For
a
function $\rho$ : $(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$, let$I_{\rho}f(x)=7n$$f(y) \frac{\rho(|x-y|)}{|x-y|^{n}}dy$.
We consider the following conditions on $\rho$:
(3.3) $\int_{0}^{1}\frac{\rho(t)}{t}dt<+\mathrm{o}\mathrm{o}$,
(3.4) $\frac{1}{A_{1}}\leq\frac{\rho(s)}{\rho(r)}\leq A_{1}$ fo$\mathrm{r}$ $\frac{1}{2}\leq\frac{s}{r}\leq 2,$
(3.5) $\frac{\rho(r)}{r^{n}}\leq A_{2}\frac{\rho(s)}{s^{n}}$ for $s\leq r.$
If$\rho(r)=r^{\alpha}$, $0<\alpha<n$, then $I_{\rho}$ is the fractional integral denoted by $I_{\alpha}$
.
Theorem 3.3. Let $/\in \mathcal{G}$, $D$,$If\in \mathcal{Y}$
.
Assume that $\phi$ is bijective.If
there exists $a$constant $A>0$ such that
$\Psi(\frac{\Phi^{-1}\circ\phi(r)\rho^{*}(r)+((\Phi^{-1}0\phi)\rho)_{*}(r)}{A})\leq\phi(r)$, $r>0,$
then
$I_{\rho}$ : $L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Psi\phi)}(\mathbb{R}^{n})$
.
If
$\Phi\in$ $\mathcal{Y}+and$ there existsa
constant $A>0$ such thatI $( \frac{(\Phi^{+})^{-1}0\phi(r)\rho^{*}(r)+(((\Phi^{+})^{-1}0\phi)\rho)_{*}(r)}{A})\leq\phi(r)$, $r>0,$
then
150
Moreover,
if
$\Phi\in \mathit{7}_{2}$, then$I_{\rho}$ : $L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$
.
Example 3.1. Let $0<\alpha<n$, $1<p$ $<q$ $<oo$, $-1/p+$$\mathrm{a}/(n-\lambda)$ $=-1/q$,
$\rho(r)=r^{\alpha}$,
$\Phi(r)=r^{p}$, $\Psi(r)=r^{q}$,
$\phi(r)=r"-n$
.
Then $\Phi\in\nabla_{2}$ and
$\Phi^{-1}05(7)\mathrm{p}’(r)$ $+((\Phi^{-1}0\phi)\rho)_{*}(r)\sim r^{(-n)/p+}$” $=r(\lambda-n)/q$.
We have
$I_{\alpha}$ : $L^{p,\lambda}(\mathbb{R}^{n})arrow L^{q,\lambda}(\mathbb{R}^{n})$
.
This is the result ofAdams [1] (1975).
Example 3.2. Let $l$ : $(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$ satisfy the doubling condition and
$\ell(r)=\{$
$(\log 1/r)^{-1}$ for small $r>0,$
$\log r$ for large $r>0.$
For $\beta>0,$ let
$\rho(r)=\{$
$(\log 1/r)^{-\beta-1}$ for small $r>0,$
$(\log r)^{\beta-1}$ for large $r>0.$
Then $\rho$ satisfies (3.3)-(3.5) and
$\rho^{*}(r)=7’$ $\frac{\rho(t)}{t}dt\sim l^{\beta}(r)$
.
Let
$\Phi(r)=r^{p}(1\leq p<\infty)$, $\Psi(r)=r^{p}\Psi^{\beta}(r)$,
and $r)=r^{\lambda-n}$ ($0\leq$ A $<n$). Then
we
have$I_{\rho}$ : $L^{1,\lambda}(\mathbb{R}^{n})=L^{(1,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Psi\phi)}(\mathbb{R}^{n})$, $I_{\rho}$ : $L^{p,\lambda}(\mathbb{R}^{n})=L^{(p,\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$,
$1<p<\infty$
.
Example 3.3. Let $\ell$ and
$\rho$ be
as
in Example 3.2. For$p>0,$ let$e_{p}(r)=\{$$1/\exp(1/r^{p})$ fo
$\mathrm{r}$ small $r>0,$
151
Let
I(r) $=e_{p}(r)$, I(r) $=e_{q}(r)$, $-1/p+\beta=-1/q$ $<0,$
$,(r)$ $=r_{:}^{-\mu}0<\mu\leq n.$
Then we have
$I_{\rho}$ : $L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$
.
Example 3.4. Let $\ell$ and
$\rho$ be
as
in Example 3.2. For $\epsilon>0$,$\delta$ $\geq 0$ and
$\mathrm{a}$ $>0,$ let
$\Phi(r)=\{$
$r(\log(1/r))^{-\epsilon-1}$ for small $r>0,$ $r(\log r)^{\delta}$ for large $r>0,$
$\Psi(r)=\{$
$r(\log(1/r))^{-\epsilon-\beta}$ for small $r>0,$
$r(\log r)^{\delta+1+\beta}$ for large $r>0,$ $6(r)$ $=r^{-\mu}$, $0<\mu\leq n.$
Then
$\Phi^{+}(r)\approx\{$
$r(\log(1/r))^{-\epsilon}$ for small $r$ $>0,$
$r(\log r)^{\delta+1}$ for large $r>0.$
Then
we
have$I_{\rho}$ : $L^{(\Phi^{+},\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$.
4. $\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{O}\mathrm{F}\mathrm{S}}$
Lemma 4.1. For $B=B(a, r)$,
$\int_{B}f(x)g(x)dx\leq 2|B|\phi(r)||f||\Phi,B,\phi||g1_{\mathrm{i},B}$
,$\phi$
.
Proofs
By Holder’s inequality for the Orlicz spaces $L^{\Phi}(B, dx/(|B|\phi(r)))$ and$L^{\tilde{\Phi}}(B, dx/(|B|\phi(r)))$,
we
have$\int_{B}f(x)g(x)\frac{dx}{|B|\phi(r)}\leq 2||f||_{L^{\Phi}(B,dx/(|B|\phi(\mathrm{r})))}||g||_{L^{\tilde{\Phi}}(B,dx/(|B|\phi(r)))}$
$=2||f||_{\Phi,B_{=}}\phi|\mathrm{K}/||\mathrm{i},B,1^{\cdot}$ $\square$
Lemma 4.2.
If
a Youngfunction
$\Phi$ is bijective and $B=B(a, r)$, then$||1||_{\tilde{\Phi},B,\phi}\leq$
152
Proof.
By $\Phi(t)s/t-\Phi(\mathrm{g})$ $\leq\Phi(t)$ for $s<t$ and $\Phi(t)s/t$ $-\Phi(s)\leq 0$ for $s\geq$ t,we
have $\tilde{\Phi}(\Phi(t)/t)\leq$
$(i).
Let $0(t)=\phi(r)$ and A $=t/\Phi(t)=$ $\mathrm{I}^{-1}$$(\phi(r))/\phi(r)$. Then$\frac{1}{|B|}\int_{B}\overline{\Phi}(\frac{1}{\lambda})dx=\tilde{\Phi}(\frac{1}{\lambda})\leq\Phi(t)=\phi(r)$
.
$\square$Lemma 4.3.
If
$f\in L^{(\Phi,\phi)}(\mathbb{R}^{n})$ and$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f\cap B(a, 2r)=l)$. Then$Mf(x)\leq C\Phi^{-1}(\phi(r))||f||_{L^{(\Phi,\phi)}}$
for
$x\in B(a, r)$.
Proof.
For all balls $B=B(b, s)$, if $s\leq r/2$ then $\int_{B}|f(x)|dx=0$, and, if $s>r/2$, then$\int_{B}|f(x)|dx\leq 2\phi(s)||f||_{\Phi,\phi,B}||1||_{\overline{\Phi},\phi,B}$
$\mathrm{E}$ $2\phi(s)||f||_{L^{(\Phi,\phi)}}!$ ”1$(\phi(s))/\phi(s)$
$\leq 2\Phi^{-1}(\phi(s))||f|6$$(+,\phi)$
$\leq C\Phi^{-1}(\phi(r))||f||_{L^{(\Phi,\phi)}}$
.
$\square$Theorem 4.4 (see [3, 10, 12]). Let $\Phi\in \mathcal{Y}$
.
Then$\sup_{t>0}\Phi(t)m(f, t)\leq c_{1}74_{n}\Phi(c_{1}|f(x)|)dx$
.
If
$\Phi\in \mathcal{Y}^{+}$, then$7_{\hslash} \Phi(Mf(x))dx\leq c_{1}\int_{\mathrm{R}^{n}}\mathrm{D}^{+}(c_{1}|f(x)|)dx$
.
If
$\Phi\in$ V2, then$\int_{\mathrm{R}^{n}}$ I$( \mathrm{f}f(x))dx\leq c_{1}\int_{\mathbb{R}^{n}}\Phi(c1|f(x)|)$ $dx$
.
Proof
of
Theorem 3.1. For all balls $B=B(a, r)$, let $f=f1+f_{2}$, $f_{1}=f\chi 2B$.
For$f_{1}$, applying Theorem 4.4,
we
have$\sup_{\mathrm{t}>0}\Phi(t)m(B, f_{1}/\lambda, t)\leq c_{1}\int_{\mathrm{R}^{n}}\Phi(c_{1}|f_{1}(x)|/\lambda)dx$, if$\Phi\in \mathcal{Y}$,
$4$$\Phi(Mf_{1}(x)/\lambda)dx\leq c_{1}4_{n}\mathrm{D}^{+}(c_{1}|f_{1}(x)|/\lambda)dx$, if I E)
$+$
We may
assume
that $c_{1}\geq 1.$ Let $c_{*}\geq 1$ be a constant such that $\phi(r)\leq c_{*}\phi(s)$ for$r\geq s.$ Let $\lambda=2^{n}c_{*}c_{1}^{2}||f||_{L}(+,\phi)$ in the cases (I) $\in \mathcal{Y}$
.
Then we have $c_{1} \int_{\mathrm{R}^{n}}D$$(c_{1}|f_{1}(x)|/ \lambda)dx=c_{1}\int_{2B}$ I$(c_{1}|f(x)|/\lambda)dx$153
Let A $=2^{n}c_{*}c_{1}^{2}||f||_{L^{(^{\mathrm{g}1+},\phi)}}$ in the
case
$\Phi\in$ )$)^{+}$. Then we have in thesame
wey$c_{1}\acute{\mathbb{R}}^{n}1^{+}(c_{1}|f_{1}(x)|/\lambda)dx\leq|B|$
$(7).
For $f_{2}$, applying Lemma 4.3, we have, for $x\in B,$
$Mf_{2}(x)\leq c_{3}\Phi^{-1}(\phi(r))||f||_{L^{(\Phi,\phi)}}$, if$\Phi\in \mathcal{Y}$,
$Mf_{2}(x)\leq c_{3}\Phi^{-1}(\phi(r))||f||_{L^{(\Phi}}+$,$), if$4!?\in$ )$)^{+}$
Let A $=c_{3}||f||\iota(\Phi,\phi)$ in the
case
$\Phi\in)^{7}$,or
A $=c_{3}||f||_{L}$($0+_{\phi)}$, in thecase
$\Phi\in y+$.
Then we have
$\sup_{t>0}\Phi(t)m(B, f_{2}/\lambda, t)\leq\int_{B}\Phi$$(Mf_{2}(x)/\lambda)dx\leq|B|\phi(r)$
.
This shows the
norm
inequations. 0Let $\Phi\in$ $\mathcal{Y}$ and $T$ is
a
operatoron
$L^{\Phi}(\mathbb{R}^{n})$ which satisfies $|7$ ($cf\mathrm{l}=|cTf|$. Ifthere exists
a
constant $c>0$ such that(4.1) $\int_{\mathrm{R}^{n}}\mathrm{D}(|Tf(x)|)dx\leq c\int_{\mathbb{R}^{n}}$ !$(c|f(x)|)dx$, $f\in L^{\Phi}(\mathbb{R}^{n})$,
then there exists a constant $C>0$ such that
(4.2) $||T$
V
$||L\Phi$ $\leq C||f||_{L^{\epsilon}}$, $f\in L^{\Phi}(\mathbb{R}^{n})$, i.e. $T:L^{\Phi}(\mathbb{R}^{n})arrow L^{\Phi}(\mathbb{R}^{n})$.If there exists
a
constant $c>0$ such that(4.3) $\sup_{t>0}CD$$(t)m(Tf, t) \leq c\int_{\mathbb{R}^{n}}\mathrm{i}$$(c|f(x)|)dx$, $f\in L$’(R
$n$
),
then there exists
a
constant $C>0$ such that(4.1) $||T$
V
$||_{L_{\mathrm{w}\cdot*\mathrm{k}}^{\Phi}}\leq C||f||_{L^{q}}$, $f\in L^{\Phi}(\mathbb{R}^{n})$, i.e. $T:L^{\Phi}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})$.If $\Phi\in$ $\Delta_{2}$, then (4.1) and (4.3) are equivalent to (4.2) and (4.4), respectively. If $T$ : $L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}(\mathbb{R}^{n})$and $T$ : $L^{p}(\mathbb{R}^{n})arrow L^{p}(\mathbb{R}^{n})$, $1<p<\infty$, then (4.1) holds
for $\Phi\in$ $\Delta_{2}\cap$
V2
and (4.3) holds for $\Phi$ $\in$ $\Delta_{2}$.
Proof
of
Theorem S.2. For $f\in L^{(\Phi,\phi)}(\mathbb{R}^{n})$ and $x\in \mathbb{R}^{n}$, we choosea
ball $B$ such that $x\in B,$ and let$Tf(x)=Tf_{1}(x)$ $+$ $7_{\mathrm{R}^{n}}K(x, y)f_{2}(y)dy$, $f=f_{1}+f_{2}$, $f_{1}=f\chi_{2B}$
.
We show that $Tf(x)$ is finite
a.e.x
and independent of the choice of $B$.
Since154
$k=1,2$, $\cdots$. Then
$\int_{\mathbb{R}^{n}}|K(x, y)f_{2}(y)|dy=\sum_{k=2}^{\infty}B_{k}\backslash B_{h-1}|K(x, y)f_{2}(y)|dy$
$\leq\sum_{k=2}^{\infty}\frac{C}{|B_{k}|}\int_{B_{k}}|f_{2}(y)|$dy.
By Lemmas 4.1 and 4.2
we
have$\frac{1}{|B_{k}|}\int_{B_{k}}|f_{2}(y)|dy\leq 2\phi(2^{k}r)||f||_{\Phi,\phi,B_{k}}||1||_{\tilde{\Phi},\phi,B_{k}}$
$\leq 2\phi(2^{k}r)||f||_{L}(\mathrm{t},\phi)$ $\mathrm{F}^{1}(\phi(2^{k}r))/\phi(2^{k}r)$
$=2\Phi^{-1}(\phi(2^{k}r))||f||_{L^{(\Phi.\phi)}}$
.
Hence
(4.5) $\int_{\mathbb{R}^{n}}|K(x, y)f_{2}(y)|dy\mathrm{S}$ $C \sum_{k=2}^{\infty}\Phi^{-1}(\phi(2^{k}r))||f||_{L^{(\Phi,\phi)}}$ $\leq C\int_{f}^{\infty})^{-1}"(\phi(t))t^{-1}dt||f||_{L}(\circ,+1$
If$x\in B\subset B’$ and $f=fi+f_{2}=f_{1}’+f_{2}’$, $f1=f\chi 2B$, $f_{1}’=f\chi 2B’$, then
$Tf(x)=Tf_{1}’(x)+ \int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$
$=T(f_{1}+f_{1}’-f_{1})(x)+ \int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$
$=$ $T(f_{1})$$(x)+T(f_{1}’-f_{1})(x)+ \int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$
$=T(f_{1})(x)+$ $/ \mathrm{R}_{n}K(x, y)(f_{1}’(y)-f_{1}(y))dy+\int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$
$=Tf_{1}(x)+ \int_{\mathrm{R}^{n}}K(x, y)f_{2}(y)dy$
.
Now we show the boundedness. For every ball $B=B(a, r)$, let $f=fi+f_{2}$, $f_{1}=$
$f\chi_{2B}$
.
For $x\in B$ we write$Tf(x)=Tf_{1}(x)+Tf_{2}(x)$, $Tf_{2}(x)= \int_{1\mathrm{R}^{\hslash}}K(x, y)f_{2}(y)dy$
.
Then, in the
case
(i), by (4.3)we
have155
and, in the
case
(ii), by (4.1)we
have$\int_{B}\Phi(\frac{|Tf_{1}(x)|}{C||f||_{L^{(\Phi,\phi)}}})dx\leq\int_{\mathbb{R}^{n}}\Phi(\frac{|Tf_{1}(x)|}{C||f||_{L^{(\Phi,\phi)}}})dx\leq C1_{n}\Phi(\frac{|f_{1}(x)|}{||f||_{L^{(\Phi.\phi)}}})dx$.
By the almost decreasingness of$\phi$ and Remark 3.1 we have
$\int_{\mathbb{R}^{n}}\Phi(\frac{|f_{1}(x)|}{||f||_{L^{(\Phi.\phi)}}})dx=\int_{2B}\Phi(\frac{|f(x)|}{||f||_{L^{(\Phi,\phi)}}})dx$
$\leq|2B|$ $6(2r)$ $\leq C|B|\phi(r)\leq C|B|\psi(r)$,
Hence
$||’ \mathrm{V}_{1}||_{\Phi,1,B,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}\leq C||f||_{L(\Phi,\phi)}$, in the
case
(i)$||Tf_{1}||_{\Phi}$,$\psi,B$ $\leq C||f||_{L}(+,\phi)$, in the
case
(ii).By (4.5)
we
have$\int_{B}\Phi(\frac{|Tf_{2}(x)|}{C||f||_{L^{(\Phi,\phi)}}})dx\leq\int_{B}\Phi(\int_{r}^{\infty}\Phi^{-1}(\phi(t))t^{-1}dtt)dx$
.
$\leq|B|\psi(r)$,i.e.
$||$
’V2H
$\Phi$,1,B,weak $\leq||7$$f_{2}||_{\Phi,1B},\leq C||f||_{L^{(\Phi,\phi)}}$
.
$\square$Proof of
Theorem 8. 3. Using the pointwise estimate in the proofof Theorem 2.2 in [17];$\Psi(\frac{|I_{\rho}f(x)|}{C_{1}||f||_{L^{(\Phi,\phi)}}})\leq\Phi(\frac{Mf(x)}{C_{0}||f||_{L^{(\Phi,\phi)}}})$ ,
and Theorem 3.1
we
have thenorm
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